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Shades and Shadows of Saddle Surface 4

Shades and Shadows of Saddle Surface 4
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2020-04-22 04:14
Solid GeometryTransformationGeometric ConstructionLocusGraphing
This example demonstrates the shades and shadows of a three-dimensional saddle surface under light, using two-dimensional geometric constructions to simulate the visual effect. The core mathematical concepts involve surfaces, projection, shadow, tangent lines, normal lines, and loci. The construction uses GeoGebra commands such as Segment, Line, Intersect, OrthogonalLine, Point, Vector, Translate, Locus, and Polygon, creating auxiliary points (A', B, C', etc.) and lines (f, g, h, etc.) to generate shadow boundaries via translation and locus. Suitable for high school or college students studying solid geometry, analytic geometry, and computer graphics, this example helps understand how 3D surfaces project onto a plane and how shadows form, serving as a visual teaching tool for mathematical visualization.

GeoGebra Commands & Parameters

  1. Segment(D, A') → f
  2. Segment(A', B) → g
  3. Line(B, f) → h
  4. Line(D, g) → i
  5. Intersect(h, i) → C'
  6. Segment(D, C') → j
  7. Segment(C', B) → k
  8. OrthogonalLine(A', g) → l
  9. OrthogonalLine(C', j) → m
  10. Point(l) → A
  11. Point(m) → C
  12. Segment(A, A') → n
  13. Segment(C, C') → p
  14. Segment(D, A) → q
  15. Segment(A, B) → r
  16. Segment(B, C) → s
  17. Segment(C, D) → t
  18. Line(D, C) → s_2
  19. Intersect(s_2, r) → E
  20. Line(E, n) → b
  21. Intersect(g, b) → G
  22. Line(B, j) → e
  23. Line(G, k) → i_1
  24. Intersect(j, i_1) → I
  25. Segment(G, I) → j_1
  26. Line(I, n) → l_1
  27. Intersect(t, l_1) → K
  28. Segment(K, I) → q_1
  29. Line(C, B) → c
  30. Intersect(c, q) → F
  31. OrthogonalLine(F, g) → g_1
  32. Intersect(g_1, f) → H
  33. Vector(A', H) → u
  34. Translate(B, u) → B'
  35. OrthogonalLine(B', g) → k_1
  36. Intersect(s, k_1) → J
  37. Segment(J, B') → m_1
  38. Segment(I, B') → n_k
  39. Point(n_k) → L
  40. Line(L, j_1) → p_1
  41. Intersect(p_1, j) → M
  42. Intersect(g, p_1) → N
  43. Intersect(p, p_1) → O
  44. Intersect(p, n) → P
  45. Line(M, n) → t_1
  46. Line(N, t_1) → a_1
  47. Intersect(t, t_1) → Q
  48. Intersect(a_1, r) → R
  49. Segment(Q, R) → c_1
  50. Line(L, t_1) → d_1
  51. Intersect(c_1, d_1) → S
  52. Locus(S, L) → ponthalmaz1
  53. Segment(D, C) → r_1
  54. Segment(C, B) → s_1
  55. Segment(A, D) → b_1
  56. Intersect(r_1, b_1) → T
  57. Intersect(r_1, s_1) → U
  58. Point(r_1) → V
  59. Segment(A, B) → e_1
  60. Polygon(A, D, U, B) → q1, a_2, d_2, u_1, b_2
  61. Point(ponthalmaz1) → W
  62. Point(ponthalmaz1) → Z
  63. Point(ponthalmaz1) → A_1
  64. Point(ponthalmaz1) → B_1
  65. Point(ponthalmaz1) → C_1
  66. Point(ponthalmaz1) → D_1
  67. Point(ponthalmaz1) → E_1
  68. Point(ponthalmaz1) → F_1
  69. Point(ponthalmaz1) → G_1
  70. Point(ponthalmaz1) → H_1
  71. Point(ponthalmaz1) → I_1
  72. Point(ponthalmaz1) → J_1
  73. Polygon(U, K, G_1, H_1, I_1, J_1, J, W, Z, A_1, B_1, C_1, D_1, E_1, F_1) → poligon1, u_2, l_2, m_2, n_2, p_2, q_2, j_3, k_2, w, z_1, f_2, g_2, h_2, i_2, j_2
  74. Segment(A', A_s) → a
  75. Intersect(b_1, e_1) → L_1
  76. Segment(L_1, A_s) → d
  77. Line(C', a) → f_1
  78. Line(U, d) → h_1
  79. Intersect(f_1, h_1) → C_s
  80. Segment(C', C_s) → n_1
  81. Segment(U, C_s) → r_2
  82. Segment(D, A_s) → t_2
  83. Segment(C_s, B) → e_2
  84. Intersect(r_1, b_1) → N_1
  85. Segment(C_s, N_1) → c_2
  86. Segment(B, A_s) → f_3
  87. Intersect(s_1, e_1) → K_1
  88. Line(K_1, C_s) → e_3
  89. Intersect(t_2, e_3) → M_1
  90. Segment(M_1, K_1) → h_4
  91. Intersect(s_1, e_1) → R_1
  92. Line(R_1, A_s) → k_7
  93. Line(A_s, N_1) → l_7
  94. Intersect(c_2, k_7) → S_8
  95. Intersect(e_2, l_7) → T_1
  96. Segment(S_8, A_s) → m_7
  97. Segment(A_s, T_1) → n_7
  98. Line(M_1, a) → i_4
  99. Intersect(f, i_4) → O_1
  100. Segment(M_1, O_1) → j_4
  101. Line(S_8, a) → p_7
  102. Line(T_1, a) → q_7
  103. Intersect(q_7, k) → U_1
  104. Intersect(p_7, j) → G_2
  105. Segment(G_2, S_8) → r_7
  106. Segment(U_1, T_1) → s_7
  107. Line(O_1, g) → k_4
  108. Intersect(k, k_4) → S_1'
  109. Segment(O_1, S_1') → l_4
  110. Line(U_1, j) → g_3
  111. Line(G_2, k) → h_3
  112. Intersect(g_3, f) → Ö_3
  113. Intersect(g, h_3) → S_2'
  114. Segment(U_1, Ö_3) → i_3
  115. Segment(S_2', G_2) → k_3
  116. Segment(S_2', S_1') → n_s
  117. Segment(Ö_3, S_2') → n_4
  118. Line(S_1', p) → m_4
  119. Intersect(s_1, m_4) → S_1
  120. Segment(S_1, S_1') → p_4
  121. Line(Ö_3, n) → l_3
  122. Line(S_2', n) → m_3
  123. Intersect(b_1, l_3) → Ö_4
  124. Intersect(e_1, m_3) → Ö_2
  125. Segment(Ö_4, Ö_3) → n_3
  126. Segment(S_2', Ö_2) → p_3
  127. Intersect(n_k, n_s) → S_k'
  128. Line(S_k', k_3) → s_4
  129. Intersect(j, s_4) → H_2
  130. Intersect(g, s_4) → I_2
  131. Segment(H_2, I_2) → t_4
  132. Line(H_2, p) → a_4
  133. Line(I_2, a_4) → b_4
  134. Intersect(r_1, a_4) → J_2
  135. Intersect(b_4, e_1) → K_2
  136. Segment(J_2, H_2) → c_4
  137. Segment(K_2, I_2) → d_4
  138. Segment(J_2, K_2) → e_4
  139. Line(S_k', n) → f_5
  140. Intersect(e_4, f_5) → S_k
  141. Segment(S_k, S_k') → g_5
  142. Line(Ö_3, a) → c_3
  143. Point(n_s) → P_1
  144. Line(P_1, k_3) → q_3
  145. Intersect(j, q_3) → Q_1
  146. Intersect(g, q_3) → V_1
  147. Line(Q_1, p) → r_3
  148. Line(V_1, p) → s_3
  149. Intersect(e_1, s_3) → W_1
  150. Intersect(r_3, r_1) → Z_1
  151. Intersect(r_3, r_1) → A_2
  152. Segment(Z_1, W_1) → t_3
  153. Line(P_1, s_3) → a_3
  154. Intersect(t_3, a_3) → B_2
  155. Locus(B_2, P_1) → ponthalmaz2
  156. Line(S_1', n_1) → b_3
  157. Line(S_2', b_3) → d_3
  158. Intersect(f_3, m_7) → C_2
  159. Intersect(d_3, f_3) → D_2
  160. Intersect(b_3, e_2) → E_2
  161. Segment(S_2', D_2) → f_4
  162. Segment(S_1', E_2) → g_4
  163. Line(P_1, g_4) → q_4
  164. Line(B_2, r_2) → r_4
  165. Intersect(q_4, r_4) → F_2
  166. Locus(F_2, P_1) → ponthalmaz3
  167. Point(s_1) → L_2
  168. Point(ponthalmaz2) → M_2
  169. Point(ponthalmaz2) → N_2
  170. Point(ponthalmaz2) → O_2
  171. Point(ponthalmaz2) → P_2
  172. Polygon(Ö_2, K_1, L_2, M_2, N_2, O_2, P_2) → poligon2, ö_2, h_5, i_5, j_5, k_5, o_2, l_5
  173. Point(ponthalmaz2) → Q_2
  174. Point(ponthalmaz2) → R_2
  175. Point(ponthalmaz2) → S_2
  176. Point(ponthalmaz2) → T_2
  177. Point(ponthalmaz2) → U_2
  178. Point(ponthalmaz2) → V_2
  179. Point(ponthalmaz2) → W_2
  180. Point(ponthalmaz2) → Z_2
  181. Point(ponthalmaz2) → A_3
  182. Point(ponthalmaz2) → B_3
  183. Point(ponthalmaz2) → C_3
  184. Point(ponthalmaz2) → D_3
  185. Point(ponthalmaz2) → E_3
  186. Point(ponthalmaz2) → F_3
  187. Point(ponthalmaz2) → G_3
  188. Point(ponthalmaz2) → H_3
  189. Point(ponthalmaz2) → I_3
  190. Point(ponthalmaz2) → J_3
  191. Point(ponthalmaz2) → K_3
  192. Point(ponthalmaz1) → L_3
  193. Point(ponthalmaz1) → M_3
  194. Point(ponthalmaz1) → N_3
  195. Point(ponthalmaz1) → O_3
  196. Point(ponthalmaz1) → P_3
  197. Point(ponthalmaz1) → Q_3
  198. Point(ponthalmaz1) → R_3
  199. Point(ponthalmaz1) → S_3
  200. Point(ponthalmaz1) → T_3
  201. Point(ponthalmaz1) → U_3
  202. Point(s_1) → V_3
  203. Polygon(S_1, Q_2, B_3, C_3, D_3, E_3, F_3, G_3, H_3, I_3, J_3, K_3, R_2, S_k, L_3, M_3, N_3, O_3, P_3, Q_3, R_3, S_3, T_3, S_2, U_3, V_3, T_2, U_2, V_2, W_2, Z_2, A_3) → poligon3, ö_1, s_5, t_5, a_5, b_5, c_5, d_5, e_5, f_6, g_6, h_6, m_5, i_6, ö_k, j_6, k_6, l_6, o_3, m_6, n_6, p_6, q_6, n_5, r_6, u_3, v_3, p_5, q_5, r_5, v_2, w_2, z_2
  204. Point(ponthalmaz1) → W_3
  205. Point(ponthalmaz1) → Z_3
  206. Point(ponthalmaz1) → A_4
  207. Point(ponthalmaz1) → B_4
  208. Point(ponthalmaz1) → C_4
  209. Point(ponthalmaz1) → D_4
  210. Point(ponthalmaz1) → E_4
  211. Point(ponthalmaz1) → F_4
  212. Point(ponthalmaz1) → G_4
  213. Point(ponthalmaz1) → H_4
  214. Point(ponthalmaz1) → I_4
  215. Point(ponthalmaz1) → J_4
  216. Point(ponthalmaz1) → K_4
  217. Point(ponthalmaz1) → L_4
  218. Point(ponthalmaz1) → M_4
  219. Point(ponthalmaz1) → N_4
  220. Point(ponthalmaz1) → O_4
  221. Point(ponthalmaz1) → P_4
  222. Point(ponthalmaz1) → Q_4
  223. Point(ponthalmaz1) → R_4
  224. Point(ponthalmaz1) → S_4
  225. Point(ponthalmaz1) → T_4
  226. Point(ponthalmaz1) → U_4
  227. Point(ponthalmaz1) → V_4
  228. Point(ponthalmaz1) → W_4
  229. Point(r_1) → Z_4
  230. Point(ponthalmaz3) → A_5
  231. Point(ponthalmaz3) → B_5
  232. Point(ponthalmaz3) → C_5
  233. Point(ponthalmaz3) → D_5
  234. Point(ponthalmaz3) → E_5
  235. Point(ponthalmaz3) → F_5
  236. Point(ponthalmaz3) → G_5
  237. Point(ponthalmaz3) → H_5
  238. Point(ponthalmaz3) → I_5
  239. Point(ponthalmaz3) → J_5
  240. Point(ponthalmaz3) → K_5
  241. Point(ponthalmaz3) → L_5
  242. Intersect(e_1, t_2) → R_6
  243. Intersect(t_2, k_6) → S_6
  244. Point(ponthalmaz1) → T_6
  245. Point(ponthalmaz1) → U_6
  246. Point(ponthalmaz1) → V_6
  247. Point(ponthalmaz1) → W_6
  248. Point(ponthalmaz1) → Z_6
  249. Point(ponthalmaz1) → A_7
  250. Point(ponthalmaz1) → B_7
  251. Point(ponthalmaz1) → C_7
  252. Point(ponthalmaz1) → D_7
  253. Point(ponthalmaz1) → E_7
  254. Point(ponthalmaz1) → F_7
  255. Point(ponthalmaz1) → G_7
  256. Point(ponthalmaz1) → H_7
  257. Point(ponthalmaz1) → I_7
  258. Intersect(i_6, ö_k) → J_7
  259. Point(ponthalmaz1) → K_7
  260. Point(ponthalmaz1) → L_7
  261. Point(ponthalmaz1) → M_7
  262. Point(ponthalmaz1) → N_7
  263. Point(ponthalmaz1) → O_7
  264. Point(ponthalmaz1) → P_7
  265. Point(ponthalmaz1) → Q_7
  266. Point(ponthalmaz1) → R_7
  267. Point(ponthalmaz1) → S_7
  268. Point(ponthalmaz1) → T_7
  269. Polygon(S_1, Q_2, C_3, D_3, E_3, G_3, H_3, J_3, K_3, J_7, W_3, Z_3, R_2, A_4, B_4, C_4, D_4, E_4, F_4, G_4, P_4, T_6, U_6, S_2, V_6, W_6, Z_6, A_7, B_7, G_7, K_7, L_7, M_7, N_7, T_2, O_7, P_7, Q_7, R_7, S_7, T_7, I_7, Z_4, U, U_2, V_2, W_2, A_3, B_3) → poligon4, s_6, h_7, i_7, j_7, t_7, a_7, b_7, c_7, d_7, e_7, w_3, t_6, z_3, f_8, g_8, h_8, i_8, j_8, k_8, l_8, m_8, n_8, a_6, u_6, v_6, w_6, z_6, p_8, q_8, r_8, s_8, t_8, a_8, b_6, b_8, o_7, c_8, d_8, e_8, f_9, g_9, h_9, z_4, u_4, c_6, d_6, e_6, f_7, g_7
  270. Point(ponthalmaz3) → M_5
  271. Point(ponthalmaz3) → N_5
  272. Point(ponthalmaz3) → O_5
  273. Point(ponthalmaz3) → P_5
  274. Point(ponthalmaz3) → Q_5
  275. Point(ponthalmaz3) → R_5
  276. Point(ponthalmaz3) → S_5
  277. Point(ponthalmaz3) → T_5
  278. Point(ponthalmaz3) → U_5
  279. Point(ponthalmaz3) → V_5
  280. Point(ponthalmaz3) → W_5
  281. Point(ponthalmaz3) → Z_5
  282. Point(ponthalmaz3) → A_6
  283. Point(ponthalmaz3) → B_6
  284. Point(ponthalmaz3) → C_6
  285. Point(ponthalmaz3) → D_6
  286. Point(ponthalmaz3) → E_6
  287. Point(ponthalmaz3) → F_6
  288. Point(t_2) → G_6
  289. Point(ponthalmaz1) → H_6
  290. Point(ponthalmaz1) → I_6
  291. Point(ponthalmaz1) → J_6
  292. Point(ponthalmaz1) → K_6
  293. Point(ponthalmaz1) → L_6
  294. Point(ponthalmaz1) → M_6
  295. Point(s_1) → N_6
  296. Point(s_1) → O_6
  297. Intersect(e_1, t_2) → P_6
  298. Polygon(A_s, D_2, U_5, V_5, W_5, Z_5, A_6, B_6, C_6, D_6, E_6, F_6, M_5, E_2, C_s, N_1, G_6, H_6, I_6, J_6, K_6, L_6, M_6, N_5, N_6, O_6, K_1, P_6, O_5, P_5, Q_5, R_5, S_5, T_5) → poligon5, a_s, q_9, u_5, v_5, w_5, z_5, r_9, s_9, t_9, a_9, b_9, i_9, c_9, d_9, c_s, e_9, f_{10}, g_{10}, h_{10}, i_{10}, j_{10}, k_{10}, j_9, l_{10}, m_{10}, o_6, n_{10}, p_{10}, k_9, o_5, l_9, m_9, n_9, p_9